246 // Spherical Sliced Bread

Spherical Sliced Bread Araminta Ponsonby took her two sets of quins to the Archimedes bakery, which makes spherical loaves. She likes to go there because each loaf is cut into ten slices, of equal thickness, so each child can have one slice of bread. They have different appetites ­ which is fortunate because some slices have smaller volume than others. But, being extremely well-behaved, all ten children love the crust, and want as much as they can get.

Which slice has the most crust?

The slices are the same

thickness: which has the

most crust?

Assume that the loaf is a perfect sphere, the slices are formed by parallel equally spaced planes, and the crust is infinitely thin ­ so the amount of crust on each slice is equal to the area of the corresponding part of the sphere's surface.

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Answer on page 309

Mathematical Theology It is said that during Leonhard Euler's second stint at the Court of Catherine the Great, the French philosopher Denis Diderot was trying to convert the Court to atheism. Since royalty generally claims to have been appointed by God, this didn't go down terribly well. At any rate, Catherine asked Euler to put a spoke in Diderot's wheel. So Euler told the Court that he knew an algebraic proof of the existence of God. Facing Diderot, he declaimed: `Sir, Mathematical Theology // 247

a þ bn =n ¼ x, hence God exists ­ reply!' Diderot had no answer, and left the Court to widespread laughter, humiliated.

Yes, well . . . There are some little problems with this anecdote, which seems to have originated with the English mathematician Augustus De Morgan in his Budget of Paradoxes. As the historian Dirk Struik pointed out in 1967, Diderot was an accomplished mathematician who had published work on geometry and probability, and would have been able to recognise nonsense when he heard it. Euler, an even better mathematician, would not have expected something that simple-minded to work. The formula is a meaningless equation unless we know what a, b, n and x are supposed to be. As Struik remarks, `No reason exists to think that the thoughtful Euler would have behaved in the asinine way indicated.'

Euler was a religious man, who apparently considered the Bible to be literal truth, but he also believed that knowledge stems, in part, from rational laws. In the eighteenth century there was occasional talk about the possibility of an algebraic proof of the Deity's existence, and Voltaire mentions one by Maupertuis in his Diatribe.

A much better attempt was found among Kurt Godel's ¨ unpublished papers. Naturally, it is formulated in terms of mathematical logic, and for the record here it is in its entirety:

Ax:1 &Vx½fðxÞ ! cðxÞ6PðfÞ ! PðcÞ

Ax:2 Pð:fÞp:PðfÞ

Th:1 PðfÞ ! }9x½fðxÞ

Df:1 GðxÞ , Vf½PðfÞ ! fðxÞ

Ax:3 PðGÞ

Th:2 }9xGðxÞ

Df:2 f ess x , fðxÞ6VCCðxÞ ! &Vx½fðxÞ ! CðxÞ

Ax:4 PðfÞ ! &PðfÞ

Th:3 GðxÞ ! G ess x

Df:3 EðxÞ , Vf½f ess x ! &9xfðxÞ

Ax:5 PðEÞ

Th:4 &9xGðxÞ

The symbolism belongs to a branch of mathematical logic called modal logic. Roughly speaking, the proof works with `positive 248 // Mathematical Theology

properties', denoted by P. The expression PðfÞ means that f is a positive property. The property `being God' is defined (Df.1) by requiring God to have all positive properties. Here G(x) means `x has the property of being God', which is a fancy way of saying `x is God'. The symbols & and } denote `necessary truth,' and `contingent truth,' respectively. The arrow ? means `implies', V is `for all' and 9 is `there exists'. The symbol : means `not', 6 is `and', and $ and , are subtly different versions of `if and only if'. The symbol `ess' is defined in Df.2. The axioms are Ax.1­5. The theorems (Th.1­4) culminate in the statement `there exists x such that x has the property of being God' ­ that is, God exists.

The distinction between necessary and contingent truth is a key novelty of modal logic. It distinguishes statements that must be true (such as `2 þ 2 ¼ 4' in a suitable axiomatic treatment of mathematics) from those that conceivably might be false (such as `it is raining today'). In conventional mathematical logic, the statement `If A then B' is always considered to be true when A is false. For instance `2 þ 2 ¼ 5 implies 1 ¼ 1' is true, and so is `2 þ 2 ¼ 5 implies 1 ¼ 42'. This may seem strange, but it is possible to prove that 1 = 1 starting from 2 þ 2 ¼ 5, and it is also possible to prove that 1 = 42 starting from 2 þ 2 ¼ 5. So the convention makes good sense. Can you find any such proofs?

If we extend this convention to human activities, then the statement `If Hitler had won World War II then Europe would now be a single nation' is trivially true, because Hitler did not win World War II. But `If Hitler had won World War II then pigs would now have wings' is also trivially true, for the same reason. In modal logic, however, it would be sensible to debate the truth or falsity of the first of these statements, depending on how history might have changed if the Nazis had won the war. The second would be false, because pigs don't have wings.

¨

Godel's sequence of statements turns out to be a formal version of the ontological argument put forward by St Anselm of Canterbury in his Proslogion of 1077­78. Defining `God' as `the greatest conceivable entity', Anselm argued that God is con- Mathematical Theology // 249

ceivable. But if he is not real, we could conceive of Him being greater by existing in reality. Therefore, God must be real.

Aside from deep issues of what we mean by `greatest' and so forth, there is a basic logical flaw here, one that every mathematician learns at his mother's knee. Before we can deduce any property of some entity or concept from its definition, we must first prove that something satisfying the definition exists. Otherwise the definition might be self-contradictory. For instance, suppose we define n to be `the largest whole number'. Then we can easily prove that n ¼ 1. For if not, n2 > n, contra- dicting the definition of n. Therefore 1 is the largest whole number. The flaw is that we cannot use any properties of n until we know that n exists. As it happens, it doesn't ­ but even if it did, we would have to prove that it did before proceeding with the deduction.

In short: in order to prove that God exists by Anselm's line of thinking, we must first establish that God exists (by some other line of reasoning, or else the logic is circular). Of course I've simplified things here, and later philosophers tried to remove the flaw by being more careful with the logic or the philosophy.

¨ Godel's proof is essentially a formal version of one proposed by

¨ Leibniz. Godel never published his proof because he was worried that it might be seen as a rigorous demonstration of the existence of God, whereas he viewed it as a formal statement of Leibniz's tacit assumptions, which would help to reveal potential logical errors. For further analysis see en.wikipedia.org/wiki/G%C3%B6del's_ontological_proof and for a detailed discussion of modal logic and its use in the proof see www.stats.uwaterloo.ca/~cgsmall/ontology1.html

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